The doctrine of double effect does not apply in this case, as I understand it. According to most accepted interpretations, double effect permits taking an action which generates negative SIDE EFFECTS, not the taking of a directly harmful action which subsequently achieve a benevolent result. In other words, its ok to perform a surgical procedure to save the life of the mother, knowing that the child will die in the process, but it's NOT permissible to kill the baby outright in order to save the mother's life.
Besides, I think you are assuming that the mother's death would AUTOMATICALLY mean the death of the child as well, which is not always the case.
You describe the proper procedure in the case of emergent danger to the mother: "to perform a surgical procedure to save the life of the mother, knowing that the child will die in the process." Under Texas law, at least, this would be defined as an abortion, a non-elective abortion.
And you are right that in the later stage of pregnancy, the child would not necessarily die if delivered.
In fact, statistically, after 15 weeks, it's safer to carry the child than to undergo intentional interventional abortion.
It's important that we make it clear that we would not demand sacrifice of her life that the mother doesn't feel prepared to make.